An optimal belief update minimising (REF ) exists when \(E_{\pi _0}(\exp (-l(\varphi , \theta ;Y,Z,\pi _0)))\)  exists,
and if this holds then the optimal valid and coherent belief update is the proper Gibbs posterior in (REF ).
The result of [1]} justifies the belief update in (REF ) for an additive loss \(l(\varphi , \theta ;Y,Z)\) . Theorem REF  below extends this to prequentially additive losses.
